# 1989 APMO Problems

## Problem 1

Let be positive real numbers, and let

.

Prove that

.

## Problem 2

Prove that the equation

has no solutions in integers except .

## Problem 3

Let be three points in the plane, and for convenience, let , . For and , suppose that is the midpoint of , and suppose that is the midpoint of . Suppose that and meet at , and that and meet at . Calculate the ratio of the area of triangle to the area of triangle .

## Problem 4

Let be a set consisting of pairs of positive integers with the property that . Show that there are at least

triples such that , , and belong to .

## Problem 5

is a strictly increasing real-valued function on reals. It has inverse . Find all possible such that for all .